1625 



Finite Summation. Recurring Series 



Finite series. Verniory, . Mathesis 12 

 (1892) 217-. 



, extension by arbitrary parameters. 

 Most, R. Arch. Mth. Ps. 48 (1868) 104-. 



-- , sum. Stern, M. A, Crelle J. 33 (1846) 

 362. 



, summation. Lindman, C. F, Stockh. 

 Ofv. 17 (1860) 415- ; Grunert Arch. 38 (1862) 

 246-. 



-- , , and application to sin n a; and cos" a;. 



Simony, O. Arch. Mth. Ps. 55 (1873) 64-. 

 Integral power of integer representable as sum 



of consecutive odd numbers. Studnicka, 



F. J. Casopis 13 (1884) 30- ; Fschr. Mth. 



(1884) 212. 

 Multinomial coefficients, sum of all. Jones, S. 



Thomson A. Ph. 4 (1822) 27-. 

 -- , summation. Masters, W. Beng. As. S. 



J. 1 (1832) 187-. 

 Numbers, theorem 2r 3 = (2r) 2 , generalisation. 



Liouville, J. Liouv. J. Mth. 2 (1857) 393-. 

 Piles of shot. Eoche, . Quetelet Cor. Mth. 



6 (1830) 199-. 



--- . Guilmin, A. N. A. Mth. 3 (1844) 



30-. 

 --- , elementary calculation. Dostor, G. 



Arch. Mth. Ps. 56 (1874) 298-. 

 --- , theory. Roche, . Gergonne A. 



Mth. 18 (1827-28) 19-. 

 -- , hexagonal. Noel, J. N. Quetelet 



Cor. Mth. 5 (1829) 347-. 



, triangular, square and oblong, formula. 

 Hubbell, H. Franklin I. J. 34 (1857) 20-. 



Polygons, sums of numbers arranged on. 



Romero, J. N. A. Mth. 1 (1882) 46-. 

 Polynomials expressing sums of powers of 



series of natural numbers. Appell, P. N. 



A. Mth. 6 (1887) 312-. 

 Powers, formation from arithmetical pro- 



gressions. Wheatstone, (Sir) C. E. S. P. 



7 (1854-55) 145-. 



, ---- . Cantor, M. N. A. Mth. 14 

 (1855) 229-. 



Product of first n numbers. Mandart, H. 



Mathesis 19 (1899) 221-. 

 --- n quantities expressed in sums of their 



powers. Day, H. G. Mess. Mth. 5 (1871) 



13. 



pansion. Schldfli, L. [1866] Crelle J. 67 

 (1867) 179-. 



H.^.a 3 ...*!*. Glaisher, J. W. L. [1877] 

 Mess. Mth. 7 (1878) 43-. 



n! expression by an integral function. 

 Hadamard, J. Ell. Sc. Mth. 19 (1895) 69-. 



Progression, arithmetical, sum of like powers 

 of terms. Ducci, E. G. Mt. 32 (1894) 348-. 



, , -- squares of terms by aid of poly- 

 gon. Romero, J. N. A. Mth. 1 (1882) 46-. 



, fractional arithmetical, summation of in- 

 tegers. Simerka, V. Casopis 5 (*1876) 39-, 

 82- ; Fschr. Mth. (*1876) 86. 



, geometrical, numbers in. Blacklewar, C. 

 E. S. P. 3 (1832) 139. 



, , 1, 3, 9, <fec. Unferdinger, F. Grunert 

 Arch. 33 (1859) 106-. 



1625 



Progressional series, summation. Benwell, 



J. B. Tilloch Ph. Mg. 58 (1821) 265-. 

 Progressions, arithmetical. Elefanti, F. E. 



S. P. 10 (1859-60) 1-. 

 , and geometrical. Pedersen, H. A. 



Mth. Ts. 1 (1859) 17-. 

 , , when 71 is fractional. Helmes, J. 



Grunert Arch. 35 (1860) 136-. 



by differences. Sardi, C. G. Mt. 11 (1873) 

 123-. 



Eational fraction, formula of Laplace's. Jur- 



gemen, C. Crelle J. 11 (1834) 136-. 

 Eecurrence, formula. Thomae, J. Crelle J. 



Mth. 110 (1892) 78-. 

 Eecurrences, system of combined. Degen, 



C. F. [1819] St. Pet. Ac. Sc. Mm. 9 



(1819-20) 130-. 

 Eecurring series. Malfatti, G. F. Verona S. 



It. Mm. 3 (1786) 571-. 

 . Svanberg, A. F. Ups. N. Acta S. Sc. 



11 (1839) 1-. 

 . Whittcorth, W. A. [1865] Mess. Mth. 



3 (1866) 117-. 

 . Ocagne, M. d'. Par. S. Mth. fill. 14 



(1886) 20- ; N. A. Mth. 9 (1890) 93- ; Par. 



EC. Pol. J. 64 (1894) 151-. 

 . Netto, E. Mh. Mth. Ps. 6 (1895) 



285-. 



an4 bilinear forms. Study, E. Mh. 

 Mth. Ps. 2 (1891) 23-. 



, conditions that scale should be irre- 

 ducible. Maillet, E. N. A. Mth. 14 (1895) 

 152-, 197-. 



connected with equation 2x 2 = 7/ 2 -l. 

 Catalan, E. C. As. Fr. C. E. 12 (1883) 

 98-. 



, elementary theory. Ocagne, M. d'. N. 



A. Mth. 3 (1884) 65-. 

 and equations. Laisant, C. A. Bll. Sc. 



Mth. As. 5 (1881) 218-. 

 , general term, and Euler's expression for 



same. Degen, C. F. [1821] Kiob. Dn. Vd. 



Selsk. Afh. 1 (1824) 135-. 

 , geometric representation. Scherk, H. F. 



Brem. Ab. 1 (1868) 225-. 

 and indeterminate analysis. Maillet, E. 



As. Fr. C. E. (1895) (Pt. 2) 233-. 

 , integration. Longchamps, G. de. As. 



Fr. C. E. (1885) (Pt. 2) 94-. 

 leading to continued fractions. Scheib- 



ner, W. Leip. B. 16 (1864) 44-. 

 , problem. Degen, C. F. [1819] St Pet. 



Ac. Sc. Mm. (1821-22) 71-. 

 , problems. Weiss, A. Crelle J. 38 (1849) 



148-. 

 for rational fractional function. Degen, 



C. F. (vi Adds.) Kiob. Ov. (1818-19) 4-. 

 , and theorem of Lagrange. Gohierre de 



Longchamps, . As. Fr. C. E. 9 (1880) 



91-. 

 , theory, addition. Vesz, J. A. (xn) 



Mag. Tud. Ak. Etk. (Mth.) 3 (1875) (No. 1) 



15pp. 



and theory of numbers. Siebeck, H. 

 Crelle J. 33 (1846) 71-. 



, type. Ocagne, M. d'. C. E. 104 (1887) 



419-, 534. 



90 



